A049009 -id:A049009 - cp's OEIS frontend

A181415 Irregular triangle a(n,k) = A049009(n,k)/n, read by rows 1<=k<=A000041(n).

1, 1, 1, 1, 6, 2, 1, 12, 9, 36, 6, 1, 20, 40, 120, 180, 240, 24, 1, 30, 75, 50, 300, 1200, 300, 1200, 2700, 1800, 120, 1, 42, 126, 210, 630, 3150, 2100, 3150, 4200, 25200, 12600, 12600, 37800, 15120, 720, 1, 56, 196, 392, 245, 1176, 7056, 11760, 8820, 11760, 11760, 88200

Offset: 1

Alford Arnold, Oct 20 2010

			Row three is calculated as follows:
( 3 18 6) divided by (3 3 3) yielding (1 6 2)
1;
1,1;
1,6,2;
1,12,9,36,6;
1,20,40,120,180,240,24;
1,30,75,50,300,1200,300,1200,2700,1800,120;
1,42,126,210,630,3150,2100,3150,4200,25200,12600,12600,37800,15120,720;

Cf. A000169 (row sums), A000081 (unlabeled rooted trees) A179438 (a similar refinement), A054589, A135278, A019538, A101817, A101818

Sum_{k=1.. A000041(n)} a(n,k) = A000169(n). (Row sums)

a(n,k) = A098546(n,k) *A049019(n,k) /n. - Compare with the formula in A101818.

Edited by R. J. Mathar, May 17 2016

A035206 Number of multisets associated with least integer of each prime signature.

Original entry on oeis.org

1, 1, 2, 1, 3, 6, 1, 4, 12, 6, 12, 1, 5, 20, 20, 30, 30, 20, 1, 6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1, 7, 42, 42, 42, 105, 210, 105, 105, 140, 420, 140, 105, 210, 42, 1, 8, 56, 56, 56, 28, 168, 336, 336, 168, 168, 280, 840, 420, 840, 70, 280, 1120, 560, 168, 420, 56, 1, 9, 72

Offset: 0

Alford Arnold

a(n,k) multiplied by A036038(n,k) yields A049009(n,k).
a(n,k) enumerates distributions of n identical objects (balls) into m of altogether n distinguishable boxes. The k-th partition of n, taken in the Abramowitz-Stegun (A-St) order, specifies the occupation of the m =m(n,k)= A036043(n,k) boxes. m = m(n,k) is the number of parts of the k-th partition of n. For the A-St ordering see pp.831-2 of the reference given in A117506. - Wolfdieter Lang, Nov 13 2007
The sequence of row lengths is p(n)= A000041(n) (partition numbers).
For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
The corresponding triangle with summed row entries which belong to partitions of the same number of parts k is A103371. [Wolfdieter Lang, Jul 11 2012]

			n\k 1  2  3  4   5   6   7   8   9  10  11  12  13 14 15
0   1
1   1
2   2  1
3   3  6  1
4   4 12  6 12   1
5   5 20 20 30  30  20   1
6   6 30 30 15  60 120  20  60  90  30   1
7   7 42 42 42 105 210 105 105 140 420 140 105 210 42  1
...
Row No. 8:  8  56 56 56 28 168 336 336 168 168 280  840 420 840 70 280 1120 560 168 420 56 1
Row No. 9: 9 72 72 72 72 252 504 504 252 252 504 84 504 1512 1512 1512 1512 504 630 2520 1260 3780 630 504 2520 1680 252 756 72 1
[rewritten and extended table by _Wolfdieter Lang_, Jul 11 2012]
a(5,5) relates to the partition (1,2^2) of n=5. Here m=3 and 5 indistinguishable (identical) balls are put into boxes b1,...,b5 with m=3 boxes occupied; one with one ball and two with two balls.
Therefore a(5,5) = binomial(5,3)*3!/(1!*2!) = 10*3 = 30. _Wolfdieter Lang_, Nov 13 2007

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Wolfdieter Lang, First 10 rows and more.

Cf. A036038, A048996, A049009.
Cf. A001700 (row sums).
Cf. A103371(n-1, m-1) (triangle obtained after summing in every row the numbers with like part numbers m).

C(sig)={my(S=Set(sig)); binomial(vecsum(sig), #sig)*(#sig)!/prod(k=1, #S, (#select(t->t==S[k], sig))!)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020

a(n,k) = A048996(n,k)*binomial(n,m(n,k)),n>=1, k=1,...,p(n) and m(n,k):=A036043(n,k) gives the number of parts of the k-th partition of n.

More terms from Joshua Zucker, Jul 27 2006

a(0)=1 prepended by Andrew Howroyd, Oct 18 2020

A019575 Place n distinguishable balls in n boxes (in n^n ways); let T(n,k) = number of ways that the maximum in any box is k, for 1 <= k <= n; sequence gives triangle of numbers T(n,k).

Original entry on oeis.org

1, 2, 2, 6, 18, 3, 24, 180, 48, 4, 120, 2100, 800, 100, 5, 720, 28800, 14700, 2250, 180, 6, 5040, 458640, 301350, 52920, 5292, 294, 7, 40320, 8361360, 6867840, 1342600, 153664, 10976, 448, 8, 362880, 172141200, 172872000, 36991080, 4644864, 387072, 20736, 648, 9

Offset: 1

Lee Corbin (lcorbin(AT)tsoft.com)

T(n,k) is the number of endofunctions on [n] such that the maximal cardinality of the nonempty preimages equals k. - Alois P. Heinz, Jul 31 2014

			Triangle begins:
       1;
       2,         2;
       6,        18,         3;
      24,       180,        48,        4;
     120,      2100,       800,      100,       5;
     720,     28800,     14700,     2250,     180,      6;
    5040,    458640,    301350,    52920,    5292,    294,     7;
   40320,   8361360,   6867840,  1342600,  153664,  10976,   448,   8;
  362880, 172141200, 172872000, 36991080, 4644864, 387072, 20736, 648, 9;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A019576. See A180281 for the case when the balls are indistinguishable.
Rows sums give A000312.
Cf. A245687.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-j, i-1, k)/j!, j=0..min(k, n))))
    end:
T:= (n, k)-> n!* (b(n$2, k) -b(n$2, k-1)):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Jul 29 2014

f[0, , b] := Boole[b == 0]; f[n_, k_, b_] := f[n, k, b] = Sum[ Binomial[b, i]*f[n - 1, k, b - i], {i, 0, Min[k, b]}]; t[n_, k_] := f[n, k, n] - f[n, k - 1, n]; Flatten[ Table[ t[n, k], {n, 1, 9}, {k, 1, n}]] (* Jean-François Alcover, Mar 09 2012, after Robert Gerbicz *)

/*setup memoization table for args <= M. Could be done dynamically inside f() */
M=10;F=vector(M,i,vector(M,i,vector(M)));
f(n,k,b)={ (!n||!b||!k) & return(!b); F[n][k][b] & return(F[n][k][b]);
F[n][k][b]=sum(i=0,min(k,b),binomial(b,i)*f(n-1,k,b-i)) }
T(n,k)=f(n,k,n)-f(n,k-1,n)
for(n=1,9,print(vector(n,k,T(n,k))))
\\ M. F. Hasler, Aug 19 2010; Based on Robert Gerbicz's code I suggest the following (very naively) memoized version of "f"

A019575(x, z) = Sum ( A049009(p)) where x = A036042(p), z = A049085(p) - Alford Arnold.
From Robert Gerbicz, Aug 19 2010: (Start)
Let f(n,k,b) = number of ways to place b balls to n boxes, where the max in any box is not larger than k. Then T(n,k) = f(n,k,n) - f(n,k-1,n). We have:
f(n, k, b) = if(n=0, if(b=0, 1, 0), Sum_{i=0..min(k, b)} binomial(b, i)*f(n-1, k, b-i)).
T(n,k) = f(n,k,n) - f(n,k-1,n). (End)

Edited by N. J. A. Sloane, Sep 06 2010

A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 6, 4, 1, 5, 10, 10, 10, 10, 5, 1, 6, 15, 15, 15, 20, 20, 20, 15, 15, 6, 1, 7, 21, 21, 21, 35, 35, 35, 35, 35, 35, 35, 21, 21, 7, 1, 8, 28, 28, 28, 28, 56, 56, 56, 56, 56, 70, 70, 70, 70, 70, 56, 56, 56, 28, 28, 8, 1, 9, 36, 36, 36, 36, 84, 84, 84, 84, 84, 84, 84

Offset: 1

Alford Arnold, Sep 14 2004

A035206 and A036038 were used to generate A049009 (Words over signatures). A098346 and A049019 provide another approach to the same end since A098346 times A049019 also yields A049009. (cf. A000312 and A000670).

Partitions are in Abramowitz and Stegun order. - Franklin T. Adams-Watters, Nov 20 2006

			A036042 begins 1 2 2 3 3 3 4 4 4 4 4 ...
A036043 begins 1 1 2 1 2 3 1 2 2 3 4 ...
so a(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...
Table begins
.
1
2 1
3 3  1
4 6  6  4  1
5 10 10 10 10 5  1
6 15 15 20 15 20 15 20 15 6 1
.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A090657, A000041 (row lengths), A098545 (row sums), A036036, A036042, A036043.

Table[Sequence @@
  Map[Function[p, Binomial[n, Length[p]]], IntegerPartitions[n]], {n,
  1, 10}] (* Olivier Gérard, May 07 2024 *)

a(n) = Combin( A036042(n), A036043(n) )

A035796 Words over signatures (derived from multisets and multinomials).

Original entry on oeis.org

1, 1, 2, 2, 3, 18, 4, 48, 6, 5, 36, 100, 144, 6, 200, 180, 600, 7, 450, 900, 294, 24, 300, 1800, 8, 882, 7200, 448, 1200, 1470, 4410, 9, 1568, 22050, 648, 7200, 3136, 1800, 9408, 10, 14700, 2592, 16200, 1960, 56448, 900, 29400, 6048, 22050, 18144

Offset: 1

Alford Arnold

nonn

A reordering of A049009(n)=A049009(p(n)): distribution of words by numeric partition where the partition sequence: p(n)=[1],[2],[1,1],[3],[2,1],[1,1,1],[4],[3,1],[2,2],[2,1,1],... (A036036) is encoded by prime factorization ([P1,P2,P3,...] with P1 >= P2 >= P3 >= ... is encoded as 2^P1 * 3^P2 * 5^P3 *...): ep(n)=2,4,6,8,12,30,16,24,36,60, ... (A036035(n)) and then sorted: s(m)=2,4,6,8,12,16,24,30,32,36,48,60,... (A025487(m)). Hence A035796(n) = A049009(s(m)).

			27 = a(5) + a(6) + a(9) since a8(4) = 3, a12(5) = 18, a30(8) = 6; 256 = a(7) + a(8) + a(11) + a(13) + a(22) = 4 + 48 + 36 + 144 + 24
27 = a(5) + a(6) + a(9) = A049009(4) + A049009(5) + A049009(6) = 3 + 18 + 6 since A036035(4) = 8 = A025487(4+1), A036035(5) = 12 = A025487(5+1), A036035(6) = 30 = A025487(8+1);...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.

Andrew Howroyd, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000312, A025487, A019575, A001700, A005651, A036035, A036036, A025487, A049009.

\\ here P is A025487 as vector and C is A049009 by partition.
GenS(lim)={my(L=List(), S=[1]); forprime(p=2, oo, listput(L, S); my(pp=vector(logint(lim, p), i, p^i)); S=concat([k*pp[1..min(if(k>1, my(f=factor(k)[, 2]); f[#f], oo), logint(lim\k, p))] | k<-S]); if(!#S, return(Set(concat(L)))) )}
P(n)={my(lim=1, v=[1]); while(#vt==S[k], sig))!) * prod(k=1, #sig, sig[k]!))}
seq(n)={[C(factor(t)[,2]) | t<-P(n)]} \\ Andrew Howroyd, Oct 18 2020

a(n) = A049009(p) where p is such that A036035(p) = A025487(n). [Corrected by Andrew Howroyd and Sean A. Irvine, Oct 18 2020]

More terms and additional comments from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 02 2001

a(1)=1 inserted by Andrew Howroyd and Sean A. Irvine, Oct 18 2020

A098545 Row sums of A098546.

Original entry on oeis.org

1, 3, 7, 21, 51, 148, 365, 983, 2461, 6360, 15687, 39757, 97033, 240425, 582622, 1421273, 3409861, 8222920, 19565707, 46680362, 110309476, 260876036, 612293443, 1437616751, 3354111156, 7823501148, 18157700800, 42112132458

Offset: 1

Alford Arnold, Sep 14 2004

By using multisets (cf. A001700) and multinomials (cf. A005651); A035206 and A036038 were used to generate A049009 (Words over signatures). A098346 and A049019 provide another approach to the same end (compare A090657).

			A098546 begins
1
1 2
1 3 3
1 4 6 6 4
so sequence begins 1 3 7 21 ...

Cf. A000041, A000312.

a(n) = Sum_{k=1..n} binomial(n, k)*A008284(n, k). - Vladeta Jovovic, Jul 24 2005

More terms from Vladeta Jovovic, Jul 24 2005

A179235 Row sums of the irregular triangle A179236.

Original entry on oeis.org

1, 4, 48, 744, 17640, 647280, 29116080

Offset: 1

Alford Arnold, Jul 04 2010

The approach used to generate A179235 and A179236 can also be used for A178802 and A049009.

			A179236 begins
1;
2,2;
6,36,6;
24,192,72,432,24;
120,1200,2400,3600,5400,4800,120;
so this sequence begins 1,4,48,...

Cf. A049009, A178802

Two more terms from R. J. Mathar, Oct 29 2011

A350788 Irregular triangle read by rows: T(n,k) is the number of partial functions on [n] such that the sizes of the preimages of the individual elements in the range form the k-th partition in the class of all partitions listed in Abramowitz and Stegun order, n>=0, 0<=k<=A000070(n).

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 2, 1, 9, 9, 18, 3, 18, 6, 1, 16, 24, 72, 16, 144, 96, 4, 48, 36, 144, 24, 1, 25, 50, 200, 50, 600, 600, 25, 400, 300, 1800, 600, 5, 100, 200, 600, 900, 1200, 120, 1, 36, 90, 450, 120, 1800, 2400, 90, 1800, 1350, 10800, 5400, 36, 900, 1800, 7200, 10800, 21600, 4320, 6, 180, 450, 1800, 300, 7200, 7200, 1800, 16200, 10800, 720

Offset: 0

Geoffrey Critzer, Jan 16 2022

The last A000041(n) entries of each row give A049009.

Row sums are (n+1)^n = A000169(n+1).

			  1,
  1,  1,
  1,  4,  2,  2,
  1,  9,  9, 18,  3,  18,  6,
  1, 16, 24, 72, 16, 144, 96, 4, 48, 36, 144, 24

Cf. A000169, A049009, A000041, A000070.

g[n_, list_] := Multinomial @@ Join[{n - Length[list]}, Table[Count[list, i], {i, 1, n}]]* Multinomial @@ Join[{n - Total[list]}, list]; Table[Map[g[nn, #] &,
   Level[Table[IntegerPartitions[k], {k, 0, nn}], {2}]], {nn, 0, 5}] // Grid

cp's OEIS Frontend

A181415 Irregular triangle a(n,k) = A049009(n,k)/n, read by rows 1<=k<=A000041(n).

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A035206 Number of multisets associated with least integer of each prime signature.

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A019575 Place n distinguishable balls in n boxes (in n^n ways); let T(n,k) = number of ways that the maximum in any box is k, for 1 <= k <= n; sequence gives triangle of numbers T(n,k).

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A098546 Table read by rows: row n has a term T(n,k) for each of the partition(n) partitions of n. T(n,k) = binomial(n,m) where m is the number of parts.

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A035796 Words over signatures (derived from multisets and multinomials).

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A098545 Row sums of A098546.

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A179235 Row sums of the irregular triangle A179236.

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A350788 Irregular triangle read by rows: T(n,k) is the number of partial functions on [n] such that the sizes of the preimages of the individual elements in the range form the k-th partition in the class of all partitions listed in Abramowitz and Stegun order, n>=0, 0<=k<=A000070(n).

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